{
  "abstract": "Let $\\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\\infty$, and let $ϕ=b/a$, where $a$ is the Pythagorean mate of $b$. In an earlier article, the author showed that, if $ϕ\\in H^2$, then \\[ f\\in\\mathcal{H}(b) \\iff \\Bigl(f\\in H^2 \\text{ and }\\sum_{m\\ge0}\\Bigl|\\sum_{n\\ge0}\\overline{\\hatϕ(n)}\\hat{f}(m+n)\\Bigr|^2<\\infty\\Bigr), \\] in which case we have \\[ \\|f\\|_{\\mathcal{H}(b)}^2=\\sum_{m\\ge0}|\\hat{f}(m)|^2 +\\sum_{m\\ge0}\\Bigl|\\sum_{n\\ge0}\\overline{\\hatϕ(n)}\\hat{f}(m+n)\\Bigr|^2. \\] It was also shown that there exists a space $\\mathcal{H}(b)$ for which $ϕ\\in\\cap_{p<2}H^p$, yet \\[ f\\in\\mathcal{H}(b)\\not\\Rightarrow \\sum_{m\\ge0}\\Bigl|\\sum_{n\\ge0}\\overline{\\hatϕ(n)}\\hat{f}(m+n)\\Bigr|^2<\\infty. \\] Here we complete this circle of ideas by showing that there exists a space $\\mathcal{H}(b)$ for which $ϕ\\in\\cap_{p<1}H^p$, yet \\[ \\Bigl(f\\in H^2\\text{ and } \\sum_{m\\ge0}\\Bigl|\\sum_{n\\ge0}\\overline{\\hatϕ(n)}\\hat{f}(m+n)\\Bigr|^2<\\infty\\Bigr) \\not\\Rightarrow f\\in\\mathcal{H}(b). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15019",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Ransford"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2qqzvz15tgksy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15019",
  "fulltext_url": null,
  "id": "2qqzvz15tgksy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15019",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7221755974",
  "pdf_url": "https://arxiv.org/pdf/2610.15019",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2qqzvz15tgksy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15019",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15019"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A counterexample concerning the coefficient formula for de Branges-Rovnyak norms",
  "title_text": "A counterexample concerning the coefficient formula for de Branges-Rovnyak norms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/2qqzvz15tgksy",
  "venue": "arXiv",
  "versions": []
}